People that regularly get haircuts are some of the worse people on earth

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Purdue Online Writing Lab says, “When refer*ring to peo*ple, both that and who can be used in infor*mal lan*guage. ‘That’ may be used to refer to the char*ac*ter*is*tics or abil*i*ties of an indi*vid*ual or a group of peo*ple.… However, when speak*ing about a par*tic*u*lar per*son in for*mal lan*guage, who is preferred.”
That said, many peo*ple and some respected ref*er*ences pre*fer “peo*ple that,” and it’s not wrong. Think Chaucer. Shakespeare. Dickens.

1 a : the person, thing, or idea indicated, mentioned, or understood from the situation b : the time, action, or event specified c : the kind or thing specified as follows d : one or a group of the indicated kind
Beware of grammar books. They very often describe the authors' opinions on what grammar should be, rather than what grammar actually is.

Specifically, I would say that it is fine to use either "that" or "who", but "who" can only be used to refer to people, while "that" refers to things and people. One should favour the word "who", if they want to clarify that it is a person, and not a thing, that they are talking about.

Jean Yates says in her book, "The Ins and Outs of Prepositions":

An adjective clause can identify a noun. The clause comes right after the noun.
...
TO IDENTIFY A PERSON, AN ADJECTIVE CLAUSE CAN BEGIN WITH WHO(M), THAT,
...
The man who(m) ...
The man that ....
...
The people who(m) ...
The people that ...
Furthermore, the "Longman Student Grammar of Spoken and Written English" by Douglas Biber, Susan Conrad, and Geoffrey Leech says:

Three relative pronouns stand out as being particularly common in English: who, which, and that.
...

That and zero are the preferred choices in conversation, although relative clauses are generally rare in that register.
Fiction is similar to conversation in its preference for that.
In contrast, news shows a much stronger preference for which and who, and academic prose strongly prefers which.
So again, I say, be wary of any single grammar book as being 100% correct. They never are.